First slide

Binomial theorem for positive integral Index

Question

Let [x] denote the greatest integer less than or equal to x. If x = $(\sqrt{3} + 1)^{5}$ then [x] is equal to

Moderate

A

75
B

50
C

76
D

152

Solution

$x = (\sqrt{3} + 1)^{5} = (\sqrt{3})^{5} +^{5} C_{1} (\sqrt{3})^{4} +^{5} C_{2} (\sqrt{3})^{3} +^{5} C_{3} (\sqrt{3})^{2} +^{5} C_{4} (\sqrt{3}) +^{5} C_{5}$
$= 9 \sqrt{3} + 45 + 30 \sqrt{3} + 30 + 5 \sqrt{3} + 1$
$= 76 + 44 \sqrt{3}$
$∴ [x] = [(\sqrt{3} + 1)^{5}]$
$\begin{array}{l} = (76 + 44 \sqrt{3}) \\ = (76) + (44 \times 1 .732) \\ = 76 + (76 .2) \\ = 76 + 76 = 152 \end{array}$

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